arXiv:1101.1777v1 [math.DS] 10 Jan 2011 PERSISTENT AND TANGENTIAL CENTER PROBLEM AND ABELIAN INTEGRALS IN DIMENSION ZERO A. ´ ALVAREZ, J.L. BRAVO, P. MARDE ˇ SI ´ C Abstract. In this paper we study the persistent and tangential cen- ter problems in dimension zero. These problems are motivated by the study of planar vector fields. The question there is when a center per- sists after deformation, or when it persists up to first order. The zero- dimensional tangential center problem is the problem of identical van- ishing of Abelian integrals on zero-dimensional cycles. Looking at the behavior of cycles at infinity, we introduce the notions of balanced, un- balanced and totally unbalanced cycles. We solve the zero-dimensional polynomial tangential center problem for totally unbalanced cycles. In [27] Pakovich and Muzychuk solved the polynomial moment prob- lem and showed that the solution is a sum of composition terms. One can say that a weak composition conjecture holds in this case. We show that the polynomial moment problem can be seen as a special case of the tangential center problem on a totally unbalanced cycle. We also study the center problem on some balanced cycles. We pro- vide examples where even the weak composition conjecture for the tan- gential center problem does not hold. In [23] Pakovich improved the result from [27] by showing that the above sum can be written with at most three terms. We show that even in the totally unbalanced case, when the weak composition conjecture holds, one cannot in general re- duce the sum to three terms. We give some applications of our results to planar tangential cen- ter problems in hyper-elliptic equations and generalized Van de Pol ’s equations. 1. Introduction This paper is inspired by the problem of characterization of the persistence of centers in deformations of Hamiltonian polynomials or rational vector fields in the plane. The problem of persistence of centers up to first order is 2010 Mathematics Subject Classification. 34C07, 34C08, 34D15, 34M35. Key words and phrases. Abelian integral, persistent center, tangential center, moment problem, center-focus problem, canard center, slow-fast system. The first author was partially supported by Junta de Extremadura and FEDER funds. The second author was partially supported by Junta de Extremadura and a MCYT/FEDER grant number MTM2008-05460. The first two authors are grateful to the Universit´ e de Bourgogne, for the hospitality and support during the visit when this work was started. The third author thanks the Universidad de Extremadura for the hospitality and support during the visit when this work was finished. 1